acsin1/4+arccos1/4

[maxo]

Въпроса ми е как се пресмята това (без калкулатор)

[dimy93]

[tex]arcsin\frac{1}{4 } =\alpha \in (\frac{\pi}{ 12} ; \frac{\pi}{ 6})[/tex] =>[tex]sin\alpha = 1/4[/tex]
[tex]arcsin\frac{1}{4 } =\beta \in ( \frac{\pi}{ 3} ; \frac{5\pi}{ 12})[/tex]=>[tex]cos\beta = 1/4[/tex]
=>[tex]sin\alpha = cos\beta[/tex] => [tex]sin( \frac{\pi}{ 2} - \beta ) = sin\alpha[/tex]
[tex]\frac{\pi}{ 2} - \beta = \alpha +2k\pi[/tex] [tex]\cup[/tex] [tex]\frac{\pi}{ 2} - \beta =\pi - \alpha +2k\pi[/tex]
[tex]\beta + \alpha \in ( \frac{5\pi}{ 12}; \frac{7\pi}{ 12})[/tex]
[tex]\alpha - \beta \in ( -\frac{\pi}{3}; -\frac{\pi}{ 6})[/tex]
=>[tex]\alpha + \beta = \frac{\pi}{2}[/tex] => [tex]arcsin\frac{1}{4 } + arcsin\frac{1}{4 } = \frac{\pi}{2}[/tex]

[mkmarinov]

[tex]arcsin(x)+arccos(x)=\frac{\pi}{2}[/tex]